All-atom Molecular Mechanics. Trent E. Balius AMS 535 / CHE /27/2010

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1 All-atom Molecular Mechanics Trent E. Balius AMS 535 / CHE /27/2010

2 Outline Molecular models Molecular mechanics Force Fields Potential energy function functional form parameters and parameterization Motivations and applications Biomolecular Force Fields

3 Molecular Models Quantum Mechanics physical, but expensive Schrödinger equation: H Ψ = E Ψ wave functions defines electron density Molecular Mechanics less physical --> empirical parameterization cheap and accurate

4

5 Molecular Mechanics Force Field Connectivity (topology) Coordinates Potential energy Parameters Potential Energy Function Functional Form V top, parm r Function of coordinates ( r ) Force Field Dependent on topology and parameters

6 Molecular Mechanics Every atom is a sphere with a radius (Lennard Jones) Point charge is located at each atomic center (Coulomb s law) Bonds and angles are held by springs to ideal lengths eg. V ( ) 2 bond = kb r r 0 Hooke's Law, K b : spring constant, r 0 : ideal length Dihedrals are represented by sigmoidal function which has energy wells at favorable angles. Improper torsions force atoms to be a defined angle to plane. Class I Potential Energy function

7 The "Tinker-toy Model" Bonded terms bonds angles dihedrals (torsions) V bond = k ( r r ) 2 b 0 V = k ( θ ) 2 angle θ = K ( 1+ cos( nχ δ )) θ 0 V dihidral χ improper V coul = q i q ε r i Through space interactions V improper ( ϕ ) 2 = k ϕ ϕ 0 R min i, = V R min LJ i = ε i, + R min 2 R min r i, i, i, = 12 ε ε * ε R min 2 ri, i i, 6

8 Dihedral Term V dihidral χ ( 1+ cos( χ δ )) = K n δ 2/n*π 2*K π (3.14) radians = 180 degrees Molecular Modelling Principles and applications, Leach Pearson Prentice hall second edition (chapter 4) Class I Potential Energy function

9 Lennard-Jones Equation V LJ = ε i, R min r i, i, 12 2 R min r i, i, 6 Energy (kcal/mol (σ, 0) (R min, ε) R min = 2 1/ 6 *σ distance (Å) Molecular Modelling Principles and applications, Leach Pearson Prentice hall second edition (chapter 4) Class I Potential Energy function

10 Potential Energy function V = bonds k b 2 2 ( b b ) + k ( θ θ ) + k ( ϕ ϕ ) 0 angles θ 0 ϕ impropers K dihedrals χ ( 1+ cos( nχ δ )) + N N ε i= 1 = i+ 1 i, R min r i, i, 12 2 R min r i, i, 6 + q i q εr i Different Force-Field CHARMM AMBER GROMOS OPLS Parameterization Experimental observables Quantum Mechanical calculations Interdependences among parameters Molecular Modelling Principles and applications, Leach Pearson Prentice hall second edition (chapter 4) Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry Class I Potential Energy function

11 Applications Dynamics of Molecules how do proteins work Energy: affinity, specificity how do proteins interact with one another and other molecules (drugs, substrates, DNA or RNA) Protein design can proteins be designed to have a specific function

12 Applications Angew. Chem. Int. Ed. 2006, 45,

13 Things to consider size of the system Energy Function sampling method Angew. Chem. Int. Ed. 2006, 45,

14 Limitations to Molecular Mechanics?

15 Limitations to MM MM cannot model the following easily: Chemical reactions Formation and breaking of bonds Polarizability Protonation states

16 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

17 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

18 Electronic Polarizability Polarization + Drude particle induced dipole models + U pol 1 = µ i E 2 µ i is the dipole moment of atom i E i is the electrostatic field at atom i i i J. Phys. Chem. B 2007, 111, Class II Potential Energy function Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

19 Electronic Polarizability Common force fields use a fixed charge model for every atom Polarizable or non-additive force fields allow for atoms to vary their charges in the context of the electric field Drude particle (Drude oscillator method) every atom has two point charges (atom center and Drude particle) connected by a spring (hooke's law)

20 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

21 Combining Rules force fields combine parameters differently arithmetic mean R min i, = R min i + R min 2 0.1, 0.9 geometric mean 0.5, 0.5 ε ε * ε i, = i A1 = A2 V LJ = ε i, R min r i, i, 12 R min r i, i, 6 ε i A1 ε ε i,, A2 ε i,,

22 Combining Rules Arithmetic mean Well depth i, Well depth Radius Radius i, Geometric mean Radius i Well depth ( ) / 2 = 0.5 ( ) / 2 = 0.5 (0.1 * 0.9)^(1 / 2) = 0.3 (0.5 * 0.5)^(1 / 2) = 0.5

23 Combining Rules Harmonic mean Euclidean relationship 2 / (1 / / 0.9) = / (1 / / 0.5) = 0.5 (0.1^ ^2)^(1 / 2) = 0.9 (0.5^ ^2)^(1 / 2) = 0.7

24 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

25 1,4 (non-bonded) Interactions 1,2 and 1,3 are normally ignored by force-fields 1,4 are scaled CHARMM no scaling AMBER and OPLS scale 0.83 Bonded terms dominate 2 3 4

26 1,5 interactions Not scaled (s = 1)

27 1,4 interactions Maybe scaled (amber: s = 0.83) Note that 1 / 1.2 = 0.83

28 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

29 Lone Pairs Force field without lone pairs do well Adding more parameters Fitting charges from quantum mechanical electrostatic potentials TIP5P and ST2 Water models

30 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

31 All-Atom vs. United Atom Force Fields Hydrogen atoms are not represented explicitly. incorporated in to the bonded heavy atom Methyl group will be represented by only one sphere Corse grain models

32 Potential Energy function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

33 Treatment of Solvation Explicit water models TIP3P, TIP4P, TIP5P, SPC,ST2 Implicit water models (Continuum with fixed dielectric) Poisson-Boltzmann (PB) model Generalized Born (GB) model This will be discussed in detail in a following class

34 Treatment of Solvation Explicit water boundaries Angew. Chem. Int. Ed. 2006, 45,

35 Potential Energy Function Electronic Polarizability Combining Rules 1,4 Interactions Lone Pairs All-Atom vs. United Atom Force Fields Treatment of Solvation Treatment of Long-Range Interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

36 Treatment of Long-Range Cut-offs Interactions Partial Mesh Ewald (PME) Smoothing functions Re-parameterization with different treatments of long-range interactions Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

37 Solvation and long range electrostatics Angew. Chem. Int. Ed. 2006, 45,

38 Biomolecular Force Fields Protein Nucleic Acid Lipid Carbohydrates Drug-Like Molecules Heterogeneous Biomolecular Systems (combining different types of biomolecules) Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

39 Conclusions Introduced Molecular Mechanics Described functional form Discussed parameterization Considered dependences and other issues Discussed applications

40 Thank you for your attention. Questions?

41 Protein Force Field Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

42 Force Field Optimization Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

43 Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

44 Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

45 Carbohydrate Force Field Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry

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